\(\log{x}=\frac{\log{2}\log{3}}{\log{4}}\)

Solve for x

The numerator is giving me issues as it has two logs in it. Can't seem to simplify it after raising both sides to 10, any tips?

Quazars
Jan 13, 2018

#1**+1 **

(If someone could help me with the Latex images just a bit bigger that'd be great, not sure why but they come up a little tiny)

I managed to get the answer myself, posting it incase anyone is curious:

Start off by changing the log from base 10 to base 2 since 2 and 4 happen to give a integer power of 2

\(\frac{\log_{2}{x}}{\log_{2}{10}}=\left( \frac{\log_{2}{2}}{\log_{2}{10}}\frac{\log_{2}{3}}{\log_{2}{10}} \right)\frac{\log_{2}{4}}{\log_{2}{10}}\)

\(\frac{\log_{2}{x}}{\log_{2}{10}}=\left( \frac{1}{\log_{2}{10}}\frac{\log_{2}{3}}{\log_{2}{10}} \frac{\log_{2}{10}}{2}\right)\)

\(\log_{2}{x}=\frac{\log_{2}{3}}{2}\)

\(x=2^{\frac{1}{2}\log_{2}{3}}\)

\(x=\sqrt{2^{\log_{2}{3}}}\)

\(x=\sqrt{3}\)

Any other methods are very welcome

Quazars
Jan 13, 2018